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Research Papers

# Singularity Conditions of 3T1R Parallel Manipulators With Identical Limb Structures

[+] Author and Article Information
Semaan Amine

Institut de Recherche en Communications et Cybernétique de Nantes, UMR CNRS n 6597, 1 rue de la Noë, 44321 Nantes, Francesemaan.amine@irccyn.ec-nantes.fr

Mehdi Tale Masouleh

Départment de génie mécanique, Université Laval, Québec, QC, G1V 0A6, Canadamehdi.tale-masouleh.1@ulaval.ca

Stéphane Caro1

Institut de Recherche en Communications et Cybernétique de Nantes, UMR CNRS n 6597, 1 rue de la Noë, 44321 Nantes, Francestephane.caro@irccyn.ec-nantes.fr

Philippe Wenger

Institut de Recherche en Communications et Cybernétique de Nantes, UMR CNRS n 6597, 1 rue de la Noë, 44321 Nantes, Francephilippe.wenger@irccyn.ec-nantes.fr

Clément Gosselin

Départment de génie mécanique, Université Laval, Québec, QC, G1V 0A6, Canadagosselin@gmc.ulaval.ca

In the scope of this paper, (s ; r ×s ) denotes a Plücker coordinate vector given by $[sr×s]$.

1

Corresponding author.

J. Mechanisms Robotics 4(1), 011011 (Feb 03, 2012) (11 pages) doi:10.1115/1.4005336 History: Received April 12, 2011; Revised September 23, 2011; Published February 03, 2012; Online February 03, 2012

## Abstract

This paper deals with the singularity analysis of parallel manipulators with identical limb structures performing Schönflies motions, namely, three independent translations and one rotation about an axis of fixed direction (3T1R). Eleven architectures obtained from a recent type synthesis of such manipulators are analyzed. The constraint analysis shows that these architectures are all overconstrained and share some common properties between the actuation and the constraint wrenches. The singularities of such manipulators are examined through the singularity analysis of the 4-R UU parallel manipulator. A wrench graph representing the constraint wrenches and the actuation forces of the manipulator is introduced to formulate its superbracket. Grassmann–Cayley Algebra is used to obtain geometric singularity conditions. Based on the concept of wrench graph, Grassmann geometry is used to show the rank deficiency of the Jacobian matrix for the singularity conditions. Finally, this paper shows the general aspect of the obtained singularity conditions and their validity for 3T1R parallel manipulators with identical limb structures.

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## Figures

Figure 8

An actuation singularity: the two actuation forces F1 and F2 are parallel

Figure 9

An actuation singularity: all actuation forces are coplanar, condition (3d) of GG

Figure 10

Wrench graph of the 4-R UU PM for the singularity conditions given in Table 3

Figure 1

3T1R PMs with identical limb structures [6]

Figure 2

The 4-R¯UU PM

Figure 3

A limb singularity, condition (a.1)

Figure 4

A limb singularity, condition (a.2)

Figure 5

Wrench graph of the 4-R UU PM in P3

Figure 6

A constraint singular configuration of the 4-R UU PM

Figure 7

A coupled motion of the 4-R UU PM

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